Hwai-chiuan Wang
Abstract:
Let
be a domain in
,
,
and
if
,
if
,
.
Consider the semilinear elliptic problem
Let
be the Sobolev space in
.
The existence, the nonexistence, and the multiplicity of positive
solutions are affected by the geometry and the topology
of the domain
.
The existence, the nonexistence, and the
multiplicity of positive solutions have been the focus of a great
deal of research in recent years.
That the above equation in a bounded domain admits a positive
solution is a classical result. Therefore the only interesting
domains in which this equation admits a positive solution are proper
unbounded domains. Such elliptic problems are difficult because of
the lack of compactness in unbounded domains. Remarkable progress
in the study of this kind of problem has been made by P. L. Lions.
He developed the concentration-compactness principles for solving
a large class of minimization problems with constraints in unbounded
domains. The characterization of domains in which this equation
admits a positive solution is an important open question.
In this monograph, we present various analyses and use them to
characterize several categories of domains in which this equation
admits a positive solution or multiple solutions.